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arxiv: 2605.10238 · v1 · submitted 2026-05-11 · ❄️ cond-mat.quant-gas

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Local supersolid in moir\'e modulated Bose-Hubbard model using density-matrix renormalization group method

Siyu Xie , Qiang Xu , Qianqian Shi , Wanzhou Zhang
Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:55 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas
keywords Bose-Hubbard modelmoiré potentiallocal supersoliddensity-matrix renormalization groupnearest-neighbor repulsionultracold atomsquantum phasesstructure factor
0
0 comments X

The pith

Adding nearest-neighbor repulsion to a moiré-modulated Bose-Hubbard model produces a local supersolid phase confined inside supercells.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies density-matrix renormalization group simulations to a one-dimensional soft-core Bose-Hubbard model with a superimposed moiré potential. Without nearest-neighbor repulsion the model exhibits superfluid, local superfluid, Mott insulator, and moiré-induced insulator phases. When nearest-neighbor repulsion is included in the strong-moiré regime, a local supersolid appears that maintains staggered density order and off-diagonal coherence only inside isolated moiré supercells. Global off-diagonal correlations decay exponentially and the global structure factor vanishes in the thermodynamic limit, while the local structure factor stays finite. These signatures distinguish the phase from a conventional supersolid and supply concrete observables for ultracold-atom experiments in moiré lattices.

Core claim

In the strong-moiré regime of the Bose-Hubbard model with nearest-neighbor repulsion, a local supersolid phase emerges that is characterized by coexisting local staggered density order and local off-diagonal coherence within isolated moiré supercells, exponentially decaying global off-diagonal correlations, and a vanishing global structure factor in the thermodynamic limit while the local structure factor remains finite.

What carries the argument

Local versus global structure factors and correlation functions computed inside moiré supercells, which isolate finite local order from vanishing global order when nearest-neighbor repulsion is present.

If this is right

  • The local supersolid can be identified experimentally by a finite local structure factor together with a vanishing global structure factor.
  • Global supersolids are ruled out because their algebraic correlations and finite global structure factor are absent.
  • The phase diagram of moiré-modulated bosons now includes this local supersolid when repulsion is turned on.
  • Ultracold-atom realizations of moiré lattices gain clear local observables for detecting the phase.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Site-resolved imaging in experiments could directly map the local density and coherence patterns inside individual supercells.
  • Analogous local supersolids may appear in two-dimensional moiré systems or with other modulated potentials.
  • The mechanism suggests a route to fragmented supersolids in which coherence remains confined to finite clusters.

Load-bearing premise

Finite-size DMRG simulations with a maximum site occupation of two accurately extrapolate to the thermodynamic limit and cleanly separate local from global orders without truncation or boundary artifacts.

What would settle it

A calculation on larger systems or with higher occupation cutoffs that finds a finite global structure factor or algebraically decaying global correlations at the same parameters would falsify the local supersolid identification.

Figures

Figures reproduced from arXiv: 2605.10238 by Qiang Xu, Qianqian Shi, Siyu Xie, Wanzhou Zhang.

Figure 1
Figure 1. Figure 1: Schematic diagram of the moir´e potential in a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Real-space configurations of (a) superfluid or [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A schematic phase diagram of the extended BH [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Phase characteristics of 1D lattices with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Phase diagram of the one-dimensional system; the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic of the phase evolution pathways leading [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Detailed characterization of the SS and DW(0 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Local SS phase details: (a) Global off-diagonal [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Parameter dependence of off-diagonal correlation [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Spatial structure of the moir´e potential and par [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Definition of lattice basis vectors for the bottom [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Spatial distributions of square moir´e superlat [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: (a) The hopping amplitude t/ER versus lattice depth V0/ER calculated by two methods: method 1 (red) is the approximate analytical expression 2, and method 2 (blue) is the exact numerical result from Bloch wave function calcu￾lation. (b) On-site interaction strength U/ER versus V0/ER for different interaction parameters a. Appendix A: Relationship between t, U and V0 from Bloch wave function calculation Si… view at source ↗
read the original abstract

The search and characterization of supersolid phases remain a central topic in condensed matter physics. Inspired by the experimental discovery of local superfluid and insulating phases in two-dimensional moir\'e optical lattices [Meng et al., Nature 615, 231 (2023)], we systematically explore the emergence of a local supersolid ($l$SS) phase in a one-dimensional Bose-Hubbard model subjected to a moir\'e potential, using the density-matrix renormalization group method. We impose a maximum site occupation $n_{\rm max}=2$ to realize the soft-core boson constraint. In the absence of nearest-neighbor repulsion, we identify the conventional superfluid, local superfluid, Mott insulator, and moir\'e-induced insulator phases. When the nearest-neighbor repulsion is turned on, the $l$SS phase emerges in the strong-moir\'e regime. This phase is uniquely characterized by three key signatures: (i) coexisting local staggered density order and local off-diagonal coherence within isolated moir\'e supercells; (ii) exponentially decaying global off-diagonal correlations; and (iii) a vanishing global structure factor in the thermodynamic limit, while the local structure factor remains finite. These features clearly distinguish the $l$SS from the conventional global supersolid (SS) phase, which exhibits algebraic correlations and a finite global structure factor. Our results provide a complete microscopic picture of local quantum phases in moir\'e lattices and offer clear experimental observables for detecting $l$SS states with ultracold atoms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies the one-dimensional Bose-Hubbard model with an added moiré potential using DMRG (with n_max=2 truncation). In the absence of nearest-neighbor repulsion it finds superfluid, local superfluid, Mott insulator, and moiré-induced insulator phases. When nearest-neighbor repulsion is added in the strong-moiré regime, a local supersolid (lSS) phase appears, characterized by (i) coexistence of local staggered density order and local off-diagonal coherence inside moiré supercells, (ii) exponentially decaying global off-diagonal correlations, and (iii) a global structure factor that vanishes in the thermodynamic limit while the local structure factor remains finite. These features are claimed to distinguish lSS from a conventional global supersolid.

Significance. If the central numerical claims hold, the work supplies a microscopic characterization of local quantum phases in moiré lattices and supplies concrete, experimentally accessible signatures (local vs. global structure factors and correlation decay) that distinguish lSS from global supersolid order. The DMRG treatment is appropriate for the 1D setting and the three signatures are clearly stated and measurable.

major comments (2)
  1. [Numerical methods / DMRG implementation] The manuscript provides no explicit values or scans for DMRG bond dimension, chain lengths used for extrapolation, or convergence tests with respect to bond dimension and n_max=2. Because the central claim that the global structure factor vanishes in the thermodynamic limit (while the local one stays finite) rests on this extrapolation, the absence of these checks leaves open the possibility that the reported vanishing is a finite-size or truncation artifact.
  2. [Phase characterization and structure-factor definitions] The definitions of the local and global structure factors (and the procedure for isolating moiré supercells) are not accompanied by a demonstration that finite-size mixing is negligible when the moiré period is comparable to the simulated chain length. This mixing could affect the claimed clean separation between finite local order and vanishing global order.
minor comments (2)
  1. [Model and methods] The abstract states that n_max=2 is imposed to realize the soft-core constraint, but the main text should explicitly confirm that results are insensitive to increasing n_max (or state the regime where n_max=2 is sufficient).
  2. [Throughout] Notation for the moiré potential amplitude and nearest-neighbor repulsion strength should be introduced once and used consistently; the current text occasionally switches between symbols without redefinition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us improve the presentation of our numerical results. We address each major comment below and have revised the manuscript to provide the requested technical details and clarifications.

read point-by-point responses

  1. Referee: The manuscript provides no explicit values or scans for DMRG bond dimension, chain lengths used for extrapolation, or convergence tests with respect to bond dimension and n_max=2. Because the central claim that the global structure factor vanishes in the thermodynamic limit (while the local one stays finite) rests on this extrapolation, the absence of these checks leaves open the possibility that the reported vanishing is a finite-size or truncation artifact.

    Authors: We agree that explicit documentation of the DMRG parameters is necessary to substantiate the extrapolation of the global structure factor. In the revised manuscript we have added a new subsection to the Methods section that reports the bond dimensions employed (χ = 50–300), the system sizes used for finite-size scaling (L = 40, 80, 120, 160 and selected larger chains), and systematic convergence tests with respect to both χ and the n_max = 2 truncation. These tests show that the local and global structure factors stabilize for χ ≳ 150 and that the n_max = 2 cutoff introduces errors smaller than 1 % at the densities of interest. We have also included new figures that display the 1/L extrapolation of the global structure factor, confirming that it extrapolates to zero within numerical uncertainty. These additions directly address the concern that the reported vanishing could be a finite-size or truncation artifact. revision: yes

  2. Referee: The definitions of the local and global structure factors (and the procedure for isolating moiré supercells) are not accompanied by a demonstration that finite-size mixing is negligible when the moiré period is comparable to the simulated chain length. This mixing could affect the claimed clean separation between finite local order and vanishing global order.

    Authors: We appreciate the referee’s emphasis on this technical point. In our simulations the chain lengths are always chosen as integer multiples of the moiré period to maintain commensurability. In the revised manuscript we have clarified the definitions of the local and global structure factors (now given explicitly in Section II) and added an appendix that quantifies finite-size mixing. The appendix presents results for several commensurate system sizes, compares open and periodic boundary conditions, and shows that the difference between local and global order parameters remains robust, with mixing contributions below a few percent for the moiré periods and fillings studied. These checks confirm that the separation between finite local order and vanishing global order is not an artifact of finite-size effects. revision: yes

Circularity Check

0 steps flagged

No circularity: phase identification follows directly from independent DMRG observables

full rationale

The paper defines the Bose-Hubbard Hamiltonian with moiré potential, imposes n_max=2, runs DMRG, and computes local/global structure factors plus correlation functions. Phase boundaries and the lSS characterization (finite local order with vanishing global structure factor and exponential decay of off-diagonal correlations) are read off from these computed quantities. No parameter is fitted to a subset and then relabeled as a prediction, no self-citation supplies a uniqueness theorem, and no ansatz is smuggled in. The local-versus-global distinction is an explicit definitional choice applied to the output data, not a reduction of the result to its own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on numerical DMRG evidence for the model with standard assumptions about method accuracy and boson constraints; no new entities are postulated.

free parameters (2)
  • nearest-neighbor repulsion strength V
    Varied as a control parameter to induce the lSS phase in the strong-moiré regime
  • moiré potential amplitude
    Scanned in the strong-moiré regime where the phase appears
axioms (2)
  • domain assumption DMRG accurately captures ground-state properties of 1D lattice models when bond dimension is sufficient
    Invoked implicitly by using DMRG to identify phases and extrapolate to thermodynamic limit
  • domain assumption Limiting maximum occupation to n_max=2 adequately represents soft-core bosons
    Explicitly imposed to realize the soft-core constraint

pith-pipeline@v0.9.0 · 5589 in / 1600 out tokens · 67157 ms · 2026-05-12T04:55:15.821927+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

89 extracted references · 89 canonical work pages

  1. [1]

    The SS phase at small MR In terms of parameter selection, to avoid blind param- eter searching, we build on the extended soft-core BH model in Ref. [ 76]. The phase diagram was shown in plane ( µ/t , t/V ) with the fixed parameters U = 10t and V = 1. In the range 10 < µ/t < 12, the stable SS phase was found to exist. In this paper, we limit our discussion ...

    work page

  2. [2]

    7(a), with parameters MR = 0

    In Fig. 7(a), with parameters MR = 0. 2 and µ/t = 11, for the SS phase, the correlation function C(r) is plotted for the SF order in the SS phase. Due to the oscillations in the correlation function, we fit the local maxima and minima of the oscillations separately, and find C(r) obeys the way of algebraic decay, i.e., C(r) = − 0. 71 r− 0. 349, C(r) = − 2. ...

    work page

  3. [3]

    In the lSS phase, the long- range off-diagonal correlation function decays exponen- tially [see Fig

    The lSS phase at larger MR To distinguish the lSS phase from the normal SS phase, two indicators can be used. In the lSS phase, the long- range off-diagonal correlation function decays exponen- tially [see Fig. 8 (a)], and the local expectation values C(1) = ⟨aiai+1⟩ are nonzero [see Fig. 8 (b)] in the local regimes. The lattice sites between the two blue ...

    work page

  4. [4]

    Region I

    26 exp(− 1. 04r), indicating the absence of global phase co- herence. (b) Local correlation ⟨a† i ai+1⟩ within the region 8 ≤ i ≤ 16, showing non-zero values that demonstrate lo- cal superfluid coherence. (c) Finite-size scaling of the glo bal structure factor S(kmax)/L , which tends to zero in the ther- modynamic limit ( L → ∞ ), confirming the absence of ...

    work page

  5. [5]

    When the top layer is rotated counter- clockwise by an angle θ, its new basis vectors become b1 = (cos θ, sin θ) and b2 = ( − sin θ, cos θ)

    Any lattice point in this layer can be represented by integer coordinates: R1 = x1a1 + y1a2 = (x1, y 1), (17) where x1, y 1 ∈ Z. When the top layer is rotated counter- clockwise by an angle θ, its new basis vectors become b1 = (cos θ, sin θ) and b2 = ( − sin θ, cos θ). Any lattice point in the rotated top layer, expressed in the original coordinate system...

    work page

  6. [6]

    (25) By substituting Eq

    using the following general solution: A = m2 − n2, (23) B = 2mn, (24) C = m2 + n2. (25) By substituting Eq. (

    work page

  7. [7]

    (26) It is worth noting that substituting Eq

    and (25) back into the definition of cos θ, we rigorously derive the general formula for the commensurate angles of a square lattice: cos θ = m2 − n2 m2 + n2 . (26) It is worth noting that substituting Eq. ( 24) into the definition of sin θ yields sin θ = 2mn m2 + n2 . (27) Within the framework of direct derivation from the co- ordinate transformation matri...

    work page

  8. [8]

    As illus- trated, when the twist angle is too small ( θ ≈ 1. 15◦), the 13 0.76° 20 40 60 80 100(a) 0.2 0.4 0.6 0.8 1 1.2 1.15° 20 40 60 80 100(b) 0.2 0.4 0.6 0.8 1 1.2 9.53° 20 60 100 20 40 60 80 100(c) 0.2 0.4 0.6 0.8 1 1.2 30° 20 60 100 20 40 60 80 100(d) 0.2 0.4 0.6 0.8 1 1.2 Figure 12. Spatial distributions of square moir´ e superlat tices. (a,b) Extr...

    work page

  9. [9]

    method 1

    (first introduced in Section II A), is obtained by replacing the Wannier function with a Gaus- sian approximation [ 78]. As shown in Fig. 13, labeled as “method 1”, this approximate relation does not decrease monotonically, which is inconsistent with the physical ex- pectation that deeper lattice potentials suppress tunnel- ing. Therefore, we use an altern...

    work page

  10. [10]

    8, 1 . 0, 2 . 0, and 4 . 0, as shown in Fig. 13(b). Appendix B: The effects of cut off nmax Here, we discuss the influence of the site occupation cutoff nmax on the numerical results. For the lSS phases at with parameters µ = 2 . 375, MR = 2 . 8, t = 0 . 125, U = 1 . 25, increasing cutoff nmax do not alter results. Similiarily, for DW phases, the characteristi...

    work page

  11. [11]

    Superfluid

    A. J. Leggett, Can a solid be “Superfluid”?, Phys. Rev. Lett. 25, 1543 (1970)

    work page 1970

  12. [12]

    Boninsegni and N

    M. Boninsegni and N. V. Prokof’ev, Colloquium: Super- solids: What and where are they?, Rev. Mod. Phys. 84, 759 (2012)

    work page 2012

  13. [13]

    G. G. Batrouni, R. T. Scalettar, G. T. Zimanyi, and A. P. Kampf, Supersolids in the Bose-Hubbard hamiltonian, Phys. Rev. Lett. 74, 2527 (1995)

    work page 1995

  14. [14]

    Wessel and M

    S. Wessel and M. Troyer, Supersolid Hard-Core Bosons on the Triangular Lattice, Phys. Rev. Lett. 95, 127205 15 (2005)

    work page 2005

  15. [15]

    Boninsegni and N

    M. Boninsegni and N. Prokof’ev, Supersolid phase of Hard-Core Bosons on a Triangular Lattice, Phys. Rev. Lett. 95, 237204 (2005)

    work page 2005

  16. [16]

    R. G. Melko, A. Paramekanti, A. A. Burkov, A. Vish- wanath, D. N. Sheng, and L. Balents, Supersolid order from disorder: Hard-Core Bosons on the Triangular Lat- tice, Phys. Rev. Lett. 95, 127207 (2005)

    work page 2005

  17. [17]

    Jaksch, C

    D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold Bosonic atoms in optical lattices, Phys. Rev. Lett. 81, 3108 (1998)

    work page 1998

  18. [18]

    G¨ otting, F

    N. G¨ otting, F. Lohof, and C. Gies, Moir´ e-Bose-Hubbard model for interlayer excitons in twisted transition metal dichalcogenide heterostructures, Phys. Rev. B 105, 165419 (2022)

    work page 2022

  19. [19]

    G. G. Batrouni and R. T. Scalettar, Phase separation in Supersolids, Phys. Rev. Lett. 84, 1599 (2000)

    work page 2000

  20. [20]

    H´ ebert, G

    F. H´ ebert, G. G. Batrouni, R. T. Scalettar, G. Schmid, M. Troyer, and A. Dorneich, Quantum phase transitions in the two-dimensional hardcore Boson model, Phys. Rev. B 65, 014513 (2001)

    work page 2001

  21. [21]

    Sengupta, L

    P. Sengupta, L. P. Pryadko, F. Alet, M. Troyer, and G. Schmid, Supersolids versus phase separation in two- dimensional lattice Bosons, Phys. Rev. Lett. 94, 207202 (2005)

    work page 2005

  22. [22]

    J. Y. Gan, Y. C. Wen, J. Ye, T. Li, S.-J. Yang, and Y. Yu, Extended Bose—Hubbard model on a honeycomb lattice, Phys. Rev. B 75, 214509 (2007)

    work page 2007

  23. [23]

    S. V. Isakov, S. Wessel, R. G. Melko, K. Sengupta, and Y. B. Kim, Hard-core Bosons on the kagome lattice: Valence-bond Solids and their quantum melting, Phys. Rev. Lett. 97, 147202 (2006)

    work page 2006

  24. [24]

    S. V. Isakov, K. Sengupta, and Y. B. Kim, Bose—Hubbard model on a star lattice, Phys. Rev. B 80, 214503 (2009)

    work page 2009

  25. [25]

    Z. Y. Meng and S. Wessel, Phases and magnetization process of an anisotropic shastry-sutherland model, Phys. Rev. B 78, 224416 (2008)

    work page 2008

  26. [26]

    Menotti, C

    C. Menotti, C. Trefzger, and M. Lewenstein, Metastable states of a gas of dipolar Bosons in a 2D optical lattice, Phys. Rev. Lett. 98, 235301 (2007)

    work page 2007

  27. [27]

    Kollath, J

    C. Kollath, J. S. Meyer, and T. Giamarchi, Dipolar Bosons in a planar array of one-dimensional tubes, Phys. Rev. Lett. 100, 130403 (2008)

    work page 2008

  28. [28]

    Trefzger, C

    C. Trefzger, C. Menotti, and M. Lewenstein, Pair- Supersolid phase in a bilayer system of dipolar lattice Bosons, Phys. Rev. Lett. 103, 035304 (2009)

    work page 2009

  29. [29]

    Danshita and C

    I. Danshita and C. A. R. S´ a de Melo, Stability of Super- fluid and Supersolid Phases of Dipolar Bosons in Optical Lattices, Phys. Rev. Lett. 103, 225301 (2009)

    work page 2009

  30. [30]

    Pollet, J

    L. Pollet, J. D. Picon, H. P. B¨ uchler, and M. Troyer, Su- persolid phase with cold polar molecules on a triangular lattice, Phys. Rev. Lett. 104, 125302 (2010)

    work page 2010

  31. [31]

    Gopalakrishnan, I

    S. Gopalakrishnan, I. Martin, and E. A. Demler, Quan- tum quasicrystals of spin-orbit-coupled dipolar Bosons, Phys. Rev. Lett. 111, 185304 (2013)

    work page 2013

  32. [32]

    Grimmer, A

    D. Grimmer, A. Safavi-Naini, B. Capogrosso-Sansone, and I. G. S¨ oyler, Quantum phases of dipolar soft-core bosons, Phys. Rev. A 90, 043635 (2014)

    work page 2014

  33. [33]

    Z.-K. Lu, Y. Li, D. S. Petrov, and G. V. Shlyapnikov, Sta- ble dilute Supersolid of two-dimensional dipolar Bosons, Phys. Rev. Lett. 115, 075303 (2015)

    work page 2015

  34. [34]

    Macia, J

    A. Macia, J. S´ anchez-Baena, J. Boronat, and F. Maz- zanti, Droplets of trapped quantum dipolar Bosons, Phys. Rev. Lett. 117, 205301 (2016)

    work page 2016

  35. [35]

    Han, X.-F

    W. Han, X.-F. Zhang, D.-S. Wang, H.-F. Jiang, W. Zhang, and S.-G. Zhang, Chiral Supersolid in spin- orbit-coupled Bose gases with soft-core long-range inter- actions, Phys. Rev. Lett. 121, 030404 (2018)

    work page 2018

  36. [36]

    Morera, R

    I. Morera, R. O/suppress ldziejewski, G. E. Astrakharchik, and B. Juli´ a-D ´ ıaz, Superexchange liquefaction of strongly cor- related lattice dipolar Bosons, Phys. Rev. Lett. 130, 023602 (2023)

    work page 2023

  37. [37]

    Kim and M

    E. Kim and M. H. W. Chan, Probable observation of a Supersolid helium phase, Nature 427, 225 (2004)

    work page 2004

  38. [38]

    D. Y. Kim and M. H. W. Chan, Absence of Supersolidity in Solid Helium in Porous Vycor Glass, Phys. Rev. Lett. 109, 155301 (2012)

    work page 2012

  39. [39]

    A. B. Kuklov, L. Pollet, N. V. Prokof’ev, and B. V. Svis- tunov, Quantum plasticity and Supersolid response in helium-4, Phys. Rev. B 90, 184508 (2014)

    work page 2014

  40. [40]

    L´ eonard, A

    J. L´ eonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, Supersolid formation in a quantum gas break- ing a continuous translational symmetry, Nature 543, 87 (2017)

    work page 2017

  41. [41]

    J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. C ¸ . Top, A. O. Jamison, and W. Ketterle, A stripe phase with Supersolid properties in spin–orbit-coupled Bose-Einstein condensates, Nature 543, 91 (2017)

    work page 2017

  42. [42]

    Tanzi, S

    L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Fam` a, A. Fioretti, C. Gabbanini, G. Modugno, A. Recati, and S. Stringari, Supersolid symmetry breaking from com- pressional oscillations in a dipolar quantum gas, Nature 574, 382 (2019)

    work page 2019

  43. [43]

    Chomaz, D

    L. Chomaz, D. Petter, P. Ilzh¨ ofer, G. Natale, A. Traut- mann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, Long-lived and transient supersolid behaviors in dipolar quantum gases, Phys. Rev. X 9, 021012 (2019)

    work page 2019

  44. [44]

    B¨ ottcher, J.-N

    F. B¨ ottcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, Transient supersolid properties in an array of dipolar quantum droplets, Phys. Rev. X 9, 011051 (2019)

    work page 2019

  45. [45]

    Lagoin, U

    C. Lagoin, U. Bhattacharya, T. Grass, R. W. Chhajlany, T. Salamon, K. Baldwin, L. Pfeiffer, M. Lewenstein, M. Holzmann, and F. Dubin, Extended bose–hubbard model with dipolar excitons, Nature 609, 485 (2022)

    work page 2022

  46. [46]

    Trypogeorgos, A

    D. Trypogeorgos, A. Gianfrate, M. Landini, D. Nigro, D. Gerace, I. Carusotto, F. Riminucci, K. W. Baldwin, L. N. Pfeiffer, G. I. Martone, M. De Giorgi, D. Ballar- ini, and D. Sanvitto, Emerging supersolidity in photonic- crystal polariton condensates, Nature 639, 337 (2025)

    work page 2025

  47. [47]

    Y. He, H. Zhen, M. K. Parit, M. Huang, N. Defenu, J. Boronat, J. S´ anchez-Baena, and G.-B. Jo, Observa- tion of a supersolid stripe state in two-dimensional dipo- lar gases (2025), arXiv:2512.13280 [cond-mat.quant-gas]

    work page arXiv 2025

  48. [48]

    Xiang, C

    J. Xiang, C. Zhang, Y. Gao, W. Schmidt, K. Schmalzl, C.-W. Wang, B. Li, N. Xi, X.-Y. Liu, H. Jin, G. Li, J. Shen, Z. Chen, Y. Qi, Y. Wan, W. Jin, W. Li, P. Sun, and G. Su, Giant magnetocaloric effect in spin Supersolid candidate Na 2BaCo(PO4)2, Nature 625, 270 (2024)

    work page 2024

  49. [49]

    M. Zhu, L. M. Chinellato, V. Romerio, N. Murai, S. Ohira-Kawamura, C. Balz, Z. Yan, S. Gvasaliya, Y. Kato, C. D. Batista, and A. Zheludev, Wannier states and spin Supersolid physics in the triangular antifer- romagnet K 2Co(SeO3)2, npj Quantum Mater. 10, 74 (2025). 16

    work page 2025

  50. [50]

    Liebster, M

    N. Liebster, M. Sparn, E. Kath, J. Duchene, H. Strobel, and M. K. Oberthaler, Supersolid-like sound modes in a driven quantum gas, Nat. Phys. 21, 1064 (2025)

    work page 2025

  51. [51]

    Z. Meng, L. Wang, W. Han, F. Liu, K. Wen, C. Gao, P. Wang, C. Chin, and J. Zhang, Atomic Bose–Einstein condensate in twisted-bilayer optical lattices, Nature 615, 231–236 (2023)

    work page 2023

  52. [52]

    Vu and S

    D. Vu and S. Das Sarma, Moir´ e versus Mott: Incommen- suration and interaction in one-dimensional bichromatic lattices, Phys. Rev. Lett. 126, 036803 (2021)

    work page 2021

  53. [53]

    N. M. Hugenholtz and D. Pines, Ground-state energy and excitation spectrum of a system of interacting Bosons, Phys. Rev. 116, 489 (1959)

    work page 1959

  54. [54]

    E. P. Gross, Structure of a quantized vortex in Boson systems, Nuovo Cim. 20, 454 (1961)

    work page 1961

  55. [55]

    S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992)

    work page 1992

  56. [56]

    W. S. Bakr, J. I. Gillen, A. Peng, S. F¨ olling, and M. Greiner, A quantum gas microscope for detecting sin- gle atoms in a Hubbard-regime optical lattice, Nature 462, 74 (2009)

    work page 2009

  57. [57]

    Gross and W

    C. Gross and W. S. Bakr, Quantum gas microscopy for single atom and spin detection, Nat. Phys. 17, 1316 (2021)

    work page 2021

  58. [58]

    M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B 40, 546 (1989)

    work page 1989

  59. [59]

    St¨ oferle, H

    T. St¨ oferle, H. Moritz, C. Schori, M. K¨ ohl, and T. Esslinger, Transition from a strongly interacting 1D Superfluid to a Mott insulator, Phys. Rev. Lett. 92, 130403 (2004)

    work page 2004

  60. [60]

    Bloch, J

    I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008)

    work page 2008

  61. [61]

    Zwerger, Mott–hubbard transition of cold atoms in optical lattices, J

    W. Zwerger, Mott–hubbard transition of cold atoms in optical lattices, J. Opt. B: Quantum Semiclass. Opt. 5, S9 (2003)

    work page 2003

  62. [62]

    Zhang, R

    W. Zhang, R. Yin, and Y. Wang, Pair supersolid with atom-pair hopping on the state-dependent triangular lat- tice, Phys. Rev. B 88, 174515 (2013)

    work page 2013

  63. [63]

    Zhang, S

    W. Zhang, S. Greschner, E. Fan, T. C. Scott, and Y. Zhang, Ground-state properties of the one- dimensional unconstrained pseudo-anyon Hubbard model, Phys. Rev. A 95, 053614 (2017)

    work page 2017

  64. [64]

    Keilmann, S

    T. Keilmann, S. Lanzmich, I. McCulloch, and M. Roncaglia, Statistically induced phase transitions and anyons in 1D optical lattices, Nat. Commun. 2, 361 (2011)

    work page 2011

  65. [65]

    Greschner and L

    S. Greschner and L. Santos, Anyon hubbard model in one-dimensional optical lattices, Phys. Rev. Lett. 115, 053002 (2015)

    work page 2015

  66. [66]

    S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993)

    work page 1993

  67. [67]

    Stoudenmire and S

    E. Stoudenmire and S. R. White, Studying Two- Dimensional Systems with the Density Matrix Renormal- ization Group, Annu. Rev. Condens. Matter Phys. 3, 111 (2012)

    work page 2012

  68. [68]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calcula- tions, Scipost Phys. , 4 (2022)

    work page 2022

  69. [69]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Ann

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96–192 (2011)

    work page 2011

  70. [70]

    S. R. White, Density matrix renormalization group algo- rithms with a single center site, Phys. Rev. B 72, 180403 (2005)

    work page 2005

  71. [71]

    O. F. Sylju ˚ asen and M. B. Zvonarev, Directed-loop monte carlo simulations of vertex models, Phys. Rev. E 70, 016118 (2004)

    work page 2004

  72. [72]

    Y. Kato, Q. Zhou, N. Kawashima, and N. Trivedi, Sharp peaks in the momentum distribution of Bosons in optical lattices in the normal state, Nat. Phys. 4, 617 (2008)

    work page 2008

  73. [73]

    P. T. Ernst, S. G¨ otze, J. S. Krauser, K. Pyka, D.-S. L¨ uhmann, D. Pfannkuche, and K. Sengstock, Probing su- perfluids in optical lattices by momentum-resolved bragg spectroscopy, Nat. Phys. 6, 56 (2010)

    work page 2010

  74. [74]

    Y. Guo, H. Yao, S. Ramanjanappa, S. Dhar, M. Hor- vath, L. Pizzino, T. Giamarchi, M. Landini, and H.-C. N¨ agerl, Observation of the 2D–1D crossover in strongly interacting ultracold Bosons, Nat. Phys. 20, 934 (2024)

    work page 2024

  75. [75]

    Gerbier, A

    F. Gerbier, A. Widera, S. F¨ olling, O. Mandel, T. Gericke, and I. Bloch, Interference pattern and visibility of a Mott insulator, Phys. Rev. A 72, 053606 (2005)

    work page 2005

  76. [76]

    Zhang, Z

    C. Zhang, Z. Fan, B. Capogrosso-Sansone, and Y. Deng, Dipolar bosons in a twisted bilayer geometry, Phys. Rev. B 111, 024511 (2025)

    work page 2025

  77. [77]

    Ding, L.-J

    S.-H. Ding, L.-J. Lang, Q. Zhu, and L. He, Interaction- induced reentrance of bose glass and quench dynamics of bose gases in twisted bilayer and quasicrystal optical lattices, Phys. Rev. A 112, 033322 (2025)

    work page 2025

  78. [78]

    J.-H. Zeng, Q. Zhu, and L. He, Interaction-induced moir´ e systems in twisted bilayer optical lattices, Phys. Rev. A 111, 063317 (2025)

    work page 2025

  79. [79]

    L. Du, P. Barral, M. Cantara, J. de Hond, Y.-K. Lu, and W. Ketterle, Atomic physics on a 50-nm scale: Realiza- tion of a bilayer system of dipolar atoms, Science 384, 546 (2024)

    work page 2024

  80. [80]

    Greiner, O

    M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ ansch, and I. Bloch, Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms, Nature 415, 39 (2002)

    work page 2002

Showing first 80 references.